Question 18·Hard·Area and Volume

A right circular cone has a height of $15$ centimeters and a base radius of $9$ centimeters. A plane parallel to the base cuts the cone, removing a smaller, similar cone from the top. The volume of the portion that remains is $\tfrac{7}{8}$ of the volume of the original cone. What is the height, in centimeters, of the smaller cone that was removed?